p.s
I forgot to specify that we're eighteen and we both enjoy cultural activities, live music, and are pretty experienced travelers.

http://www.mcgill.ca/students/housing/summer
Congratulations on your graduation, happy trip planning and welcome to Fodors. Sometimes universities offer summer housing for budget travelers. It looks like McGill university offers such housing in numerous downtown locations. Above is the link to summer housing offers.
Good luck to you!

University de Quebec a Montreal (UQAM) also rents out its residences during the summer. The eastern building (Residence Est, on Rene-Levesque) is handy for the jazz festival, a couple of blocks from a major subway station and only two stops to the festival site. It's also on the edge of the Latin Quarter nightlife. Main concern on Trip Advisor is lack of air conditioning. Here is the site in French http://www.residences-uqam.qc.ca/hotel/ You will be served at reception in English if you need it. Some travel reservations sites such as Expedia will show you information in English although I'm not sure reservations are available yet. Telephone may be the best bet.
The jazz festival fills the downtown hotels so you need to reserve as soon as it is possible. Many events are free but the ticketed concerts also sell fast. Some shows are already available. June 28-July 7; you can watch the calendar at http://www.montrealjazzfest.com/default-en.aspx

You will need to book rooms early, as the Jazz Festival attracts lots of people - it's the largest jazz festival in the world. Being on a budget, the jazz fest is pefect since there are all day long free outdoor concerts in the downtown, often several at once. As far as inexpensive dining, the BYOB restaurants on the Plateau are lots of fun. There is a large concentration of them on Duluth Street, just off St. Denis. Many have outdoor terraces, and there is everything from Afghan to Italian. Because of the jazz fest, the city will be very lively both night and day. There are plenty of hip clubs throughout the city if you want to get away from the crowds at the jazz fest. St. Denis is lined with outdoor cafes, clubs, shops, etc, etc. Other inexpensive activities would be exploring Jean Talon Market, the Notre Dame Basillica, and renting a bike at one of the many rental stations and biking the hundreds of miles of trails that cris-cross the city.
As far as Montral neighborhoods - the most interesting are the Plateau, Old Montreal, the downtown, and the Mile End. All these areas are quite safe, and great for strolling around.
Quebec City is quite a bit smaller, but also very interesting, with lots of French charm. There's always lots of activity on the prominade, and the views are fabulous. If possible stay in or near the old city in Quebec City.